Growth and covering theorems associated with the Roper-Suffridge extension operator

Abstract

The Roper-Suffridge extension operator, originally introduced in the context of convex functions, provides a way of extending a (locally) univalent function f ∈ Hol ⁡ ( D , C ) f\in \operatorname {Hol}(\mathbb {D},\mathbb {C}) to a (locally) univalent map F ∈ Hol ⁡ ( B n , C n ) F\in \operatorname {Hol}(B_{n},\mathbb {C}^{n}) . If f f belongs to a class of univalent functions which satisfy a growth theorem and a distortion theorem, we show that F F satisfies a growth theorem and consequently a covering theorem. We also obtain covering theorems of Bloch type: If f f is convex, then the image of F F (which, as shown by Roper and Suffridge, is convex) contains a ball of radius π / 4 \pi /4 . If f ∈ S f\in S , the image of F F contains a ball of radius 1 / 2 1/2 .